The potential communication capacity of optical fibers operating in the low loss wavelength windows of 1.3 .mu.m and 1.5 .mu.m is in the order of tens of Terahertz. The practical utilization of this bandwidth my be realized through the use of wavelength division multiplexing (WDM), in which the spectral range is subdivided and allocated to different carrier wavelengths (channels) which are multiplexed onto the same fiber. The frequency bandwidth that an individual channel occupies depends on a number of factors, including the impressed modulation bandwidth, margins to accommodate for carrier frequency drift, carrier frequency uncertainty, and to reduce cross-talk between channels.
Although an isolated optical fiber may inherently have tremendous information carrying capacity, the overall optical communication link may be significantly restricted in bandwidth. These restrictions may result from the limited optical amplifier spectral windows, the availability of lasing sources and their tuning ranges, and filter tuning ranges. Hence, to achieve efficient use of bandwidth requires that the available communications windows be densely filled with multiplexed channels. At the input and output of such a system, filters are needed to combine and separate wavelengths in individual channels. The performance of these wavelength filters, in their ability to filter one channel and reject out of band signals, is critical in determining channel spacing and hence channel density for WDM communications.
An ideal filter response is a "box" or "window" function, characterized by unity filtering of the wavelength band of interest, and zero transmission of signals outside of the band. The bandwidth of each channel ideally is made as narrow as possible, to accommodate a larger number of channels.
Filters may be subdivided into two very broad categories, reflection type and transmission type.
In the reflection type the wavelength of interest is coupled from a forward travelling wave into a backward travelling wave, i.e. is reflected back in the direction from which it came, either into the same waveguide or into another waveguide. Particular types of reflection filters include, for example, a distributed Bragg reflector (DBR), a distributed feedback (DFB) reflector, and a contra-directional coupler.
The transmission type of filters on the other hand include for example a co-directional coupler, grating assisted coupler, and Mach Zehnder filter, which couple energy between two waves or modes travelling in the same directions.
Filters operate through a wavelength dependent exchange of power between two waveguide modes. It is well known that two waveguides placed in close proximity may exchange power through their evanescent fields, which penetrate the guiding layer of the other waveguide. This power exchange occurs continuously along the propagation direction, with a rate determined by the inter-waveguide spacing, and the degree of velocity matching of the two modes.
For co-directional couplers comprising two parallel waveguides, and for the case of a uniform coupling coefficient along the interaction length, the transfer function is approximately given by a [sin(.chi.)/.chi.].sup.2 function, with symmetric first sidelobes of -9 dB. This level of sidelobe suppression allows for significant cross talk between channels and provides unacceptable wavelength selectivity for current communications applications.
Codirectional couplers are modeled conventionally through a set of coupled mode differential equations written in general form, such as: ##EQU2## where A.sub.1 and A.sub.2 represent the amplitudes of the two waveguide modes at position z in the coupler. .DELTA..beta. is the detuning constant and K is the coupling strength. K depends on the waveguide structure and is strongly influenced by the separation of adjacent waveguides. The origin of coupling may be due to interaction of evanescent fields in a codirectional coupler, or by coherent scattering in a grating assisted coupler. By varying the interaction strength K along the coupler the spectral response of the device can be improved. Thus by a suitable choice of function K(z) it is theoretically possible to generate a desired filter response. The calculation of the taper shape required for a desired response has been a long-standing design question.
For example, an original proposal was based on an approximate Fourier transform relation, described by Alferness et al., in IEEE J. Quantum Electronics QE 14(11) pp. 843-847, November 1978. Improved optical waveguide directional couplers were suggested for which the coupling strength is weighted, or tapered, along the interaction length by several known taper functions. For example, in two co-directional waveguides, the inter-guide separation may be varied along the interaction length. In an article in Appl. Phys. Letters 35(3), pp. 260-263, 1 Aug. 1979, Alferness demonstrated experimentally the feasibility of using weighted coupling to reduce sidelobes of the filter transfer response, allowing for closer stacking of wavelength channels with reduced cross talk by use of various taper profiles. In particular, a Hamming taper function was found to provide -25 dB transfer response sidelobes, a significant improvement over other known taper functions, for example, a raised cosine function, Blackman taper, and Kaiser taper function.
The goal of filter synthesis is to solve for the coupling constant function K(z) given a desired response for A.sub.1 and A.sub.2. However, when K(z) is non-constant, the set of coupled equations (1a) and (1b) has no analytic solution in general. Hence, filter design is currently guided by a set of approximate solutions.
The most important of these approximate solutions is obtained by the Fourier transform relation, given by: EQU A.sub.2 (.DELTA..beta.).apprxeq..intg.K(z)e.sup.-j.DELTA..beta..z dzEquation (2)
In Equation (2), A.sub.2 (.DELTA..beta.) is the amplitude in the output or coupled waveguide as a function of detuning .DELTA..beta., (which may be related to the actual wavelength .lambda.). Because Equation (2) represents a Fourier transform relation between K(z) in the spatial domain and A.sub.2 (.DELTA..beta.) in the wavelength domain, the principle of duality may be used. That is, given a desired A.sub.2 (.DELTA..beta.), K(z) is found by the inverse Fourier transform. This approximation is valid for small coupling values, and does not extend to describe the critical region of the main passband and first few sidelobes. No analytic solution currently treats the important region around the central wavelength.
On the other hand, inverse scattering methods are mathematically rigorous. Given a desired response A.sub.2 (.DELTA..beta.), the inverse scattering method attempts to numerically solve the coupled mode equations in an inverse sense, thus yielding the desired interaction function K(z). The success of the inversion depends on being able to specify the desired response as a rational function, and is therefore limited to those functions which fall under this category. Moreover, the interaction strength solution is defined on the entire z axis {-.infin.&lt;z&lt;.infin.}. Thus one must arbitrarily truncate the range of K(z) to get a coupler of finite length. This truncation can seriously degrade performance, as discussed in an article of Song et al., entitled "Design of corrugated waveguide filters by the Gel'fand-Levitan-Machenko inverse scattering method." in J. Opt. Soc. Am. A. Vol.2 (11), pp. 1905-1915, 1985.
The inverse scattering method does not yield any guidelines on how to obtain the desired response in a coupler with a specified and finite length. Thus, it can never, in a rigorous sense, yield the ideal solution of K(z) for a practical device. Indeed, no known finite taper function K(z) has yet been shown to meet the required optical communication specifications. Thus alternative approaches are required for filter design.
Consequently, as described in the above mentioned co-pending U.S. patent application Ser. No. 08/385,419 to the current inventors, entitled "Taper shapes for ultra-low sidelobe levels in directional coupler filters", a novel approach to filter design was developed based on a variational optimization theory and a publication in Optics Letters, vol. 20 (11) pp. 1259-1261, 1 Jun. 1995. By this approach, a new class of coupler shape functions was synthesized and their application was demonstrated in deriving taper functions for co-directional coupler filters, in which: EQU .kappa.(z)=L.sub.o (z)+SL.sub.1 (z)+S.sup.2 L.sub.2 (z)+S.sup.3 L.sub.3 (z)+S.sup.4 L.sub.4 (z) Equation (3)
where .kappa.(z) is the normalized interaction strength, S is the desired sidelobe level, and where L.sub.i are given by ##EQU3## and where for a real physical device, the physical length scales as Z=sL.sub.c. where Z is the physical length, z is the normalized length, and L.sub.c is the length of the interaction region of the device. Thus L.sub.i are functions of the propagation distance z only, It was demonstrated that sidelobe suppression of the transfer function from -40 dB to -75 dB can be obtained, with bandwidths within 5% of the theoretical minimum bandwidths, may be obtained for co-directional couplers.
Another type of coupled waveguide filters are known as grating assisted codirectional couplers is discussed in detail in another copending patent application to the present inventors, entitled "Taper shapes for sidelobe suppression and bandwidth minimization in distributed feedback optical reflection filters", to be filed concurrently herewith. In these devices, power exchange occurs due to coherent scattering by a periodic grating placed in proximity to the two waveguides. In an article by Sakata, on wavelength selective grating assisted couplers, in Optics Letters, Vol. 17(7) 1 Apr. 1992 pp. 463-465, improved sidelobe suppression was obtained by controlling the duty ratio of grating assisted vertically coupled waveguides, using a truncated Gaussian taper function. Sidelobe suppression in grating assisted wavelength selective couplers of Sakata showed that required grating periods were sufficiently coarse (.about..mu.m) to allow for fabrication by known conventional photolithography and etching process. It was also shown that a Hamming taper offered narrower bandwidth spacing, while a Kaiser taper provided improved sidelobe suppression below -60 dB. Thus, in comparing the Hamming, Blackman, Kaiser, and truncated Gaussian taper functions, the trade off between multiplexing density and cross-talk level was demonstrated.
Nevertheless, reflection filters exhibit distinctly different types of wavelength filter response, or spectral response, from transmission filters.
Reflection filters are characterized by having a much narrower bandwidth than transmission filters, which is an advantage for densely spaced optical channels. A reflection filter is also characterized by what is called a "stopband", that is a wavelength range or window where all wavelengths exhibit strong reflection, and are stopped from being transmitted through the device. Thus the spectral response of a reflection filter is closer to an ideal `box` function. On the other hand, reflection filters tend to have much larger sidelobes outside the stopband range of wavelengths, which are troublesome for cross talk. Sidelobe suppression to a level of at least -30 dB to -60 dB is desirable for current communications applications, to reduce cross-talk to an acceptable level. Improved taper functions for reflection filters are required to provide the desired level of sidelobe suppression with a minimum bandwidth.